Toniolo and Linder, Equation (7)

Percentage Accurate: 33.4% → 85.1%

Time: 23.1s

Alternatives: 11

Speedup: 225.0×

Specification

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 33.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \end{array} \]

FPCore

(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))

C

double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Fortran

real(8) function code(x, l, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: l
    real(8), intent (in) :: t
    code = (sqrt(2.0d0) * t) / sqrt(((((x + 1.0d0) / (x - 1.0d0)) * ((l * l) + (2.0d0 * (t * t)))) - (l * l)))
end function

Java

public static double code(double x, double l, double t) {
	return (Math.sqrt(2.0) * t) / Math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
}

Python

def code(x, l, t):
	return (math.sqrt(2.0) * t) / math.sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)))

Julia

function code(x, l, t)
	return Float64(Float64(sqrt(2.0) * t) / sqrt(Float64(Float64(Float64(Float64(x + 1.0) / Float64(x - 1.0)) * Float64(Float64(l * l) + Float64(2.0 * Float64(t * t)))) - Float64(l * l))))
end

MATLAB

function tmp = code(x, l, t)
	tmp = (sqrt(2.0) * t) / sqrt(((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l)));
end

Wolfram

code[x_, l_, t_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * t), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(x + 1.0), $MachinePrecision] / N[(x - 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] + N[(2.0 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(l * l), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \sqrt{2}\\ t_3 := \frac{\sqrt{2} \cdot t\_m}{t\_2 + 0.5 \cdot \frac{2 \cdot \left(2 \cdot \left(t\_m \cdot t\_m\right) + l\_m \cdot l\_m\right)}{x \cdot t\_2}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-186}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 5 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{l\_m}\\ \mathbf{elif}\;t\_m \leq 1.02 \cdot 10^{-142}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_m \leq 2.8 \cdot 10^{+69}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{\left(t\_m \cdot t\_m\right) \cdot \left(\left(2 + \frac{2}{x}\right) + 2 \cdot \left(\frac{\frac{l\_m}{x} \cdot l\_m}{t\_m \cdot t\_m} + \frac{1}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \end{array} \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (let* ((t_2 (* t_m (sqrt 2.0)))
        (t_3
         (/
          (* (sqrt 2.0) t_m)
          (+
           t_2
           (*
            0.5
            (/ (* 2.0 (+ (* 2.0 (* t_m t_m)) (* l_m l_m))) (* x t_2)))))))
   (*
    t_s
    (if (<= t_m 6.2e-186)
      t_3
      (if (<= t_m 5e-174)
        (/ (* (sqrt x) t_m) l_m)
        (if (<= t_m 1.02e-142)
          t_3
          (if (<= t_m 2.8e+69)
            (*
             t_m
             (sqrt
              (/
               2.0
               (*
                (* t_m t_m)
                (+
                 (+ 2.0 (/ 2.0 x))
                 (* 2.0 (+ (/ (* (/ l_m x) l_m) (* t_m t_m)) (/ 1.0 x))))))))
            (pow (/ (+ x 1.0) (+ x -1.0)) -0.5))))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * sqrt(2.0);
	double t_3 = (sqrt(2.0) * t_m) / (t_2 + (0.5 * ((2.0 * ((2.0 * (t_m * t_m)) + (l_m * l_m))) / (x * t_2))));
	double tmp;
	if (t_m <= 6.2e-186) {
		tmp = t_3;
	} else if (t_m <= 5e-174) {
		tmp = (sqrt(x) * t_m) / l_m;
	} else if (t_m <= 1.02e-142) {
		tmp = t_3;
	} else if (t_m <= 2.8e+69) {
		tmp = t_m * sqrt((2.0 / ((t_m * t_m) * ((2.0 + (2.0 / x)) + (2.0 * ((((l_m / x) * l_m) / (t_m * t_m)) + (1.0 / x)))))));
	} else {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * sqrt(2.0d0)
    t_3 = (sqrt(2.0d0) * t_m) / (t_2 + (0.5d0 * ((2.0d0 * ((2.0d0 * (t_m * t_m)) + (l_m * l_m))) / (x * t_2))))
    if (t_m <= 6.2d-186) then
        tmp = t_3
    else if (t_m <= 5d-174) then
        tmp = (sqrt(x) * t_m) / l_m
    else if (t_m <= 1.02d-142) then
        tmp = t_3
    else if (t_m <= 2.8d+69) then
        tmp = t_m * sqrt((2.0d0 / ((t_m * t_m) * ((2.0d0 + (2.0d0 / x)) + (2.0d0 * ((((l_m / x) * l_m) / (t_m * t_m)) + (1.0d0 / x)))))))
    else
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double t_2 = t_m * Math.sqrt(2.0);
	double t_3 = (Math.sqrt(2.0) * t_m) / (t_2 + (0.5 * ((2.0 * ((2.0 * (t_m * t_m)) + (l_m * l_m))) / (x * t_2))));
	double tmp;
	if (t_m <= 6.2e-186) {
		tmp = t_3;
	} else if (t_m <= 5e-174) {
		tmp = (Math.sqrt(x) * t_m) / l_m;
	} else if (t_m <= 1.02e-142) {
		tmp = t_3;
	} else if (t_m <= 2.8e+69) {
		tmp = t_m * Math.sqrt((2.0 / ((t_m * t_m) * ((2.0 + (2.0 / x)) + (2.0 * ((((l_m / x) * l_m) / (t_m * t_m)) + (1.0 / x)))))));
	} else {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	t_2 = t_m * math.sqrt(2.0)
	t_3 = (math.sqrt(2.0) * t_m) / (t_2 + (0.5 * ((2.0 * ((2.0 * (t_m * t_m)) + (l_m * l_m))) / (x * t_2))))
	tmp = 0
	if t_m <= 6.2e-186:
		tmp = t_3
	elif t_m <= 5e-174:
		tmp = (math.sqrt(x) * t_m) / l_m
	elif t_m <= 1.02e-142:
		tmp = t_3
	elif t_m <= 2.8e+69:
		tmp = t_m * math.sqrt((2.0 / ((t_m * t_m) * ((2.0 + (2.0 / x)) + (2.0 * ((((l_m / x) * l_m) / (t_m * t_m)) + (1.0 / x)))))))
	else:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	t_2 = Float64(t_m * sqrt(2.0))
	t_3 = Float64(Float64(sqrt(2.0) * t_m) / Float64(t_2 + Float64(0.5 * Float64(Float64(2.0 * Float64(Float64(2.0 * Float64(t_m * t_m)) + Float64(l_m * l_m))) / Float64(x * t_2)))))
	tmp = 0.0
	if (t_m <= 6.2e-186)
		tmp = t_3;
	elseif (t_m <= 5e-174)
		tmp = Float64(Float64(sqrt(x) * t_m) / l_m);
	elseif (t_m <= 1.02e-142)
		tmp = t_3;
	elseif (t_m <= 2.8e+69)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(t_m * t_m) * Float64(Float64(2.0 + Float64(2.0 / x)) + Float64(2.0 * Float64(Float64(Float64(Float64(l_m / x) * l_m) / Float64(t_m * t_m)) + Float64(1.0 / x))))))));
	else
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	t_2 = t_m * sqrt(2.0);
	t_3 = (sqrt(2.0) * t_m) / (t_2 + (0.5 * ((2.0 * ((2.0 * (t_m * t_m)) + (l_m * l_m))) / (x * t_2))));
	tmp = 0.0;
	if (t_m <= 6.2e-186)
		tmp = t_3;
	elseif (t_m <= 5e-174)
		tmp = (sqrt(x) * t_m) / l_m;
	elseif (t_m <= 1.02e-142)
		tmp = t_3;
	elseif (t_m <= 2.8e+69)
		tmp = t_m * sqrt((2.0 / ((t_m * t_m) * ((2.0 + (2.0 / x)) + (2.0 * ((((l_m / x) * l_m) / (t_m * t_m)) + (1.0 / x)))))));
	else
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(t$95$2 + N[(0.5 * N[(N[(2.0 * N[(N[(2.0 * N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(l$95$m * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.2e-186], t$95$3, If[LessEqual[t$95$m, 5e-174], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision], If[LessEqual[t$95$m, 1.02e-142], t$95$3, If[LessEqual[t$95$m, 2.8e+69], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(t$95$m * t$95$m), $MachinePrecision] * N[(N[(2.0 + N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(N[(N[(N[(l$95$m / x), $MachinePrecision] * l$95$m), $MachinePrecision] / N[(t$95$m * t$95$m), $MachinePrecision]), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 6.20000000000000018e-186 or 5.0000000000000002e-174 < t < 1.0200000000000001e-142

   1. Initial program 37.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   if 6.20000000000000018e-186 < t < 5.0000000000000002e-174

   1. Initial program 1.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]

   if 1.0200000000000001e-142 < t < 2.79999999999999982e69

   1. Initial program 47.9%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]

   if 2.79999999999999982e69 < t

   1. Initial program 26.5%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+138}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+304}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{2 \cdot \left(t\_m \cdot \left(t\_m + \frac{t\_m}{x}\right)\right) + \frac{l\_m \cdot \left(l\_m \cdot 2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t\_m}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot l\_m\right)}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e+138)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (if (<= (* l_m l_m) 5e+304)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+ (* 2.0 (* t_m (+ t_m (/ t_m x)))) (/ (* l_m (* l_m 2.0)) x)))))
      (/ (* (sqrt 2.0) t_m) (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l_m)))))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+138) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else if ((l_m * l_m) <= 5e+304) {
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	} else {
		tmp = (sqrt(2.0) * t_m) / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 5d+138) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else if ((l_m * l_m) <= 5d+304) then
        tmp = t_m * sqrt((2.0d0 / ((2.0d0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0d0)) / x))))
    else
        tmp = (sqrt(2.0d0) * t_m) / (sqrt((1.0d0 / x)) * (sqrt(2.0d0) * l_m))
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+138) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else if ((l_m * l_m) <= 5e+304) {
		tmp = t_m * Math.sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	} else {
		tmp = (Math.sqrt(2.0) * t_m) / (Math.sqrt((1.0 / x)) * (Math.sqrt(2.0) * l_m));
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 5e+138:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	elif (l_m * l_m) <= 5e+304:
		tmp = t_m * math.sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))))
	else:
		tmp = (math.sqrt(2.0) * t_m) / (math.sqrt((1.0 / x)) * (math.sqrt(2.0) * l_m))
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e+138)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	elseif (Float64(l_m * l_m) <= 5e+304)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(t_m * Float64(t_m + Float64(t_m / x)))) + Float64(Float64(l_m * Float64(l_m * 2.0)) / x)))));
	else
		tmp = Float64(Float64(sqrt(2.0) * t_m) / Float64(sqrt(Float64(1.0 / x)) * Float64(sqrt(2.0) * l_m)));
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e+138)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	elseif ((l_m * l_m) <= 5e+304)
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	else
		tmp = (sqrt(2.0) * t_m) / (sqrt((1.0 / x)) * (sqrt(2.0) * l_m));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+138], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+304], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(t$95$m * N[(t$95$m + N[(t$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * N[(l$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] * t$95$m), $MachinePrecision] / N[(N[Sqrt[N[(1.0 / x), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * l$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.00000000000000016e138

   1. Initial program 47.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 5.00000000000000016e138 < (*.f64 l l) < 4.9999999999999997e304

   1. Initial program 16.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 4.9999999999999997e304 < (*.f64 l l)

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   6. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   8. Simplified 0

      \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 79.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+138}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{elif}\;l\_m \cdot l\_m \leq 5 \cdot 10^{+269}:\\ \;\;\;\;t\_m \cdot \sqrt{\frac{2}{2 \cdot \left(t\_m \cdot \left(t\_m + \frac{t\_m}{x}\right)\right) + \frac{l\_m \cdot \left(l\_m \cdot 2\right)}{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{l\_m} \cdot t\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= (* l_m l_m) 5e+138)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (if (<= (* l_m l_m) 5e+269)
      (*
       t_m
       (sqrt
        (/
         2.0
         (+ (* 2.0 (* t_m (+ t_m (/ t_m x)))) (/ (* l_m (* l_m 2.0)) x)))))
      (* (/ (sqrt x) l_m) t_m)))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+138) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else if ((l_m * l_m) <= 5e+269) {
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	} else {
		tmp = (sqrt(x) / l_m) * t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((l_m * l_m) <= 5d+138) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else if ((l_m * l_m) <= 5d+269) then
        tmp = t_m * sqrt((2.0d0 / ((2.0d0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0d0)) / x))))
    else
        tmp = (sqrt(x) / l_m) * t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if ((l_m * l_m) <= 5e+138) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else if ((l_m * l_m) <= 5e+269) {
		tmp = t_m * Math.sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	} else {
		tmp = (Math.sqrt(x) / l_m) * t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if (l_m * l_m) <= 5e+138:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	elif (l_m * l_m) <= 5e+269:
		tmp = t_m * math.sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))))
	else:
		tmp = (math.sqrt(x) / l_m) * t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (Float64(l_m * l_m) <= 5e+138)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	elseif (Float64(l_m * l_m) <= 5e+269)
		tmp = Float64(t_m * sqrt(Float64(2.0 / Float64(Float64(2.0 * Float64(t_m * Float64(t_m + Float64(t_m / x)))) + Float64(Float64(l_m * Float64(l_m * 2.0)) / x)))));
	else
		tmp = Float64(Float64(sqrt(x) / l_m) * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if ((l_m * l_m) <= 5e+138)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	elseif ((l_m * l_m) <= 5e+269)
		tmp = t_m * sqrt((2.0 / ((2.0 * (t_m * (t_m + (t_m / x)))) + ((l_m * (l_m * 2.0)) / x))));
	else
		tmp = (sqrt(x) / l_m) * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+138], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], If[LessEqual[N[(l$95$m * l$95$m), $MachinePrecision], 5e+269], N[(t$95$m * N[Sqrt[N[(2.0 / N[(N[(2.0 * N[(t$95$m * N[(t$95$m + N[(t$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(l$95$m * N[(l$95$m * 2.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 5.00000000000000016e138

   1. Initial program 47.8%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 5.00000000000000016e138 < (*.f64 l l) < 5.0000000000000002e269

   1. Initial program 19.3%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in l around inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 5.0000000000000002e269 < (*.f64 l l)

   1. Initial program 0.2%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 1.08 \cdot 10^{+167}:\\ \;\;\;\;{\left(\frac{x + 1}{x + -1}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 1.08e+167)
    (pow (/ (+ x 1.0) (+ x -1.0)) -0.5)
    (/ (* (sqrt x) t_m) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.08e+167) {
		tmp = pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = (sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 1.08d+167) then
        tmp = ((x + 1.0d0) / (x + (-1.0d0))) ** (-0.5d0)
    else
        tmp = (sqrt(x) * t_m) / l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 1.08e+167) {
		tmp = Math.pow(((x + 1.0) / (x + -1.0)), -0.5);
	} else {
		tmp = (Math.sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 1.08e+167:
		tmp = math.pow(((x + 1.0) / (x + -1.0)), -0.5)
	else:
		tmp = (math.sqrt(x) * t_m) / l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 1.08e+167)
		tmp = Float64(Float64(x + 1.0) / Float64(x + -1.0)) ^ -0.5;
	else
		tmp = Float64(Float64(sqrt(x) * t_m) / l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 1.08e+167)
		tmp = ((x + 1.0) / (x + -1.0)) ^ -0.5;
	else
		tmp = (sqrt(x) * t_m) / l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 1.08e+167], N[Power[N[(N[(x + 1.0), $MachinePrecision] / N[(x + -1.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 1.08000000000000005e167

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 1.08000000000000005e167 < l

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 80.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 4.4 \cdot 10^{+167}:\\ \;\;\;\;\sqrt{\frac{x + -1}{x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 4.4e+167)
    (sqrt (/ (+ x -1.0) (+ x 1.0)))
    (/ (* (sqrt x) t_m) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 4.4e+167) {
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 4.4d+167) then
        tmp = sqrt(((x + (-1.0d0)) / (x + 1.0d0)))
    else
        tmp = (sqrt(x) * t_m) / l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 4.4e+167) {
		tmp = Math.sqrt(((x + -1.0) / (x + 1.0)));
	} else {
		tmp = (Math.sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 4.4e+167:
		tmp = math.sqrt(((x + -1.0) / (x + 1.0)))
	else:
		tmp = (math.sqrt(x) * t_m) / l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 4.4e+167)
		tmp = sqrt(Float64(Float64(x + -1.0) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(sqrt(x) * t_m) / l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 4.4e+167)
		tmp = sqrt(((x + -1.0) / (x + 1.0)));
	else
		tmp = (sqrt(x) * t_m) / l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 4.4e+167], N[Sqrt[N[(N[(x + -1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 4.40000000000000007e167

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   if 4.40000000000000007e167 < l

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 7.2 \cdot 10^{+167}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x} \cdot t\_m}{l\_m}\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 7.2e+167)
    (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
    (/ (* (sqrt x) t_m) l_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 7.2e+167) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else {
		tmp = (sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 7.2d+167) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else
        tmp = (sqrt(x) * t_m) / l_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 7.2e+167) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else {
		tmp = (Math.sqrt(x) * t_m) / l_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 7.2e+167:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	else:
		tmp = (math.sqrt(x) * t_m) / l_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 7.2e+167)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	else
		tmp = Float64(Float64(sqrt(x) * t_m) / l_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 7.2e+167)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	else
		tmp = (sqrt(x) * t_m) / l_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 7.2e+167], N[(1.0 - N[(N[(1.0 - N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] * t$95$m), $MachinePrecision] / l$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 7.20000000000000049e167

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 7.20000000000000049e167 < l

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 80.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;l\_m \leq 3.5 \cdot 10^{+167}:\\ \;\;\;\;1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{x}}{l\_m} \cdot t\_m\\ \end{array} \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (*
  t_s
  (if (<= l_m 3.5e+167)
    (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))
    (* (/ (sqrt x) l_m) t_m))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+167) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else {
		tmp = (sqrt(x) / l_m) * t_m;
	}
	return t_s * tmp;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (l_m <= 3.5d+167) then
        tmp = 1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x)
    else
        tmp = (sqrt(x) / l_m) * t_m
    end if
    code = t_s * tmp
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	double tmp;
	if (l_m <= 3.5e+167) {
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	} else {
		tmp = (Math.sqrt(x) / l_m) * t_m;
	}
	return t_s * tmp;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	tmp = 0
	if l_m <= 3.5e+167:
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x)
	else:
		tmp = (math.sqrt(x) / l_m) * t_m
	return t_s * tmp

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	tmp = 0.0
	if (l_m <= 3.5e+167)
		tmp = Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x));
	else
		tmp = Float64(Float64(sqrt(x) / l_m) * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, l_m, t_m)
	tmp = 0.0;
	if (l_m <= 3.5e+167)
		tmp = 1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x);
	else
		tmp = (sqrt(x) / l_m) * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * If[LessEqual[l$95$m, 3.5e+167], N[(1.0 - N[(N[(1.0 - N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[x], $MachinePrecision] / l$95$m), $MachinePrecision] * t$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if l < 3.49999999999999987e167

   1. Initial program 39.6%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in x around -inf 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   if 3.49999999999999987e167 < l

   1. Initial program 0.0%

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 0

      \[\leadsto expr\]

   5. Simplified 0

      \[\leadsto expr\]

   7. Applied egg-rr 0

      \[\leadsto expr\]

   8. Taylor expanded in t around 0 0

      \[\leadsto expr\]

   10. Simplified 0

       \[\leadsto expr\]

   12. Applied egg-rr 0

       \[\leadsto expr\]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.6% accurate, 17.3× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 - \frac{1 - \frac{0.5 + \frac{-0.5}{x}}{x}}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (- 1.0 (/ (- 1.0 (/ (+ 0.5 (/ -0.5 x)) x)) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 - ((1.0d0 - ((0.5d0 + ((-0.5d0) / x)) / x)) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 - Float64(Float64(1.0 - Float64(Float64(0.5 + Float64(-0.5 / x)) / x)) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 - ((1.0 - ((0.5 + (-0.5 / x)) / x)) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 - N[(N[(1.0 - N[(N[(0.5 + N[(-0.5 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

7. Applied egg-rr 0

   \[\leadsto expr\]

8. Taylor expanded in x around -inf 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 9: 76.5% accurate, 25.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 - \frac{\frac{-0.5}{x} + 1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m)
 :precision binary64
 (* t_s (- 1.0 (/ (+ (/ -0.5 x) 1.0) x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - (((-0.5 / x) + 1.0) / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 - ((((-0.5d0) / x) + 1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 - (((-0.5 / x) + 1.0) / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 - (((-0.5 / x) + 1.0) / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 - Float64(Float64(Float64(-0.5 / x) + 1.0) / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 - (((-0.5 / x) + 1.0) / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 - N[(N[(N[(-0.5 / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

7. Applied egg-rr 0

   \[\leadsto expr\]

8. Taylor expanded in x around -inf 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 10: 76.3% accurate, 45.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(1 + \frac{-1}{x}\right) \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s (+ 1.0 (/ -1.0 x))))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * (1.0d0 + ((-1.0d0) / x))
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * (1.0 + (-1.0 / x));
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * (1.0 + (-1.0 / x))

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * Float64(1.0 + Float64(-1.0 / x)))
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * (1.0 + (-1.0 / x));
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in l around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in x around inf 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 11: 75.7% accurate, 225.0× speedup

Math

\[\begin{array}{l} l_m = \left|\ell\right| \\ t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot 1 \end{array} \]

FPCore

l_m = (fabs.f64 l)
t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x l_m t_m) :precision binary64 (* t_s 1.0))

C

l_m = fabs(l);
t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Fortran

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, l_m, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: l_m
    real(8), intent (in) :: t_m
    code = t_s * 1.0d0
end function

Java

l_m = Math.abs(l);
t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double l_m, double t_m) {
	return t_s * 1.0;
}

Python

l_m = math.fabs(l)
t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, l_m, t_m):
	return t_s * 1.0

Julia

l_m = abs(l)
t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, l_m, t_m)
	return Float64(t_s * 1.0)
end

MATLAB

l_m = abs(l);
t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, l_m, t_m)
	tmp = t_s * 1.0;
end

Wolfram

l_m = N[Abs[l], $MachinePrecision]
t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, l$95$m_, t$95$m_] := N[(t$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 36.3%

   \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

7. Applied egg-rr 0

   \[\leadsto expr\]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024110 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))